Instructions
Go to the basement floor of the library in the mansion section of the asylum. You will see the Joker's face appear on a television screen and receive the objective "Rescue the Hostages." The Joker will tell you the hostages will die if you do not rescue them within 2 minutes.
Walk back up the stairs behind you to get to the main floor of the library. The library has a main floor and three floors above it with open areas in the center. Use the grappling hook to go to the second floor and walk around until you see a grate near the ground. There is a stairway on this level, but it is blocked so ignore it.
Remove the grate and enter the vent. It has a few side paths, but they are dead ends. Keep crawling until you reach an open room. Walk to the opposite end and use the grappling hook to climb up to the open vent on the wall. You will pass through another vent and into another area with another vent high on the wall that ends in a grate.
Remove the grate and you will arrive on the top floor with a giant chandelier straight ahead. Cut the chain holding the chandelier with a batarang. The chandelier will fall, crashing through the floor below and freeing the hostages. Jump over the railing and glide down to the basement floor.